We begin with the case of an automorphism \(\varphi \)  of order 2, which corresponds to the case of hyperelliptic surfaces; such an automorphism leaves all spin structures invariant. By Lemma REF  (see also Example REF ), the lift \(\tilde{\varphi }\)  has order 4, and (after possibly taking the opposite lift) it acts on the space of harmonic spinors as multiplication by \(i\) . Via Example REF , Proposition REF  is then a direct consequence of the computation of the \(h^0(L)\)  for the corresponding linear systems, which can be found in [1]} and [2]}; the statement is the following.
